On Fibonacci numbers with few prime divisors

If $n$ is a positive integer, write $F_n$ for the $n$th Fibonacci number, and $\omega(n)$ for the number of distinct prime divisors of $n$. We give a description of Fibonacci numbers satisfying $\omega(F_n) \leq 2$. Moreover, we prove that the inequality $\omega(F_n) \geq (\log n)^{\log 2 + o(1)}$ holds for almost all $n$. We conjecture that $\omega(F_n) \gg \log n$ for composite $n$, and give a heuristic argument in support of this conjecture.